Music and Mathematics Part 1

The past year or so seems to have hosted a steady trickle of articles, blog posts and public debates about the connection (or, more usually, the lack thereof) between music and maths.

Discussions involving mathematics bring on a sense of alienation/torpor to many in the general public, but I’m one of those who find mathematical thinking exciting, exacting, exhilarating. And as music is (really quite literally) on my mind all the time, I am deeply interested in the assertions of others regarding the links (or lack thereof) between these two (musical and mathematical) aspects of organisational thought and expression.

It turns out that many of those who spend time disputing the existence of links between music and mathematics go on to reveal that they were never that good at maths. In fact, they confess that they’ve failed key mathematical assessments throughout their schooling. What they don’t acknowledge is that they have a vested interest in denying connections between mathematical and musical thinking, and no one seems to think it worth mentioning that someone who is no good at maths probably won’t have a very nuanced idea as to what mathematics actually is [and therefore is probably not best equipped to detail how unconnected music and maths might be].

Along these lines, if you think that mathematics is just a fancy word for “counting” the argument will go something like this: maths and music are linked because in music you do counting (of beats and intervals). That’s it. As soon as we notice that music is more than counting (either beats or intervals) it’s no great stretch to be convinced the hoopla about maths and music having all that much in common must be based on a trite understanding of what music is.

The problem is that the trite understanding isn’t of music so much as it is of mathematics. Mathematical thinking does involve quantity (which concept does involve, amongst many other things, “counting”), but it also necessarily involves spatial thinking/awareness as well as pattern recognition, two substantial non-counting aspects of how to think in and about the world.

But even within the confines of “quantity” we find ourselves in a world of relationship: “this is bigger than that” might not sound particularly profound, but a lot about our experience of music can be described within this single concept of quantity. No matter which way we hear it, louder, longer, faster, further, more (and their corresponding softer, shorter, slower, closer, less) describe nearly everything that can happen in music.

Stephen Hough makes a quite convincing case that it is the ambiguities of music that make it wildly different to mathematics, that mathematics is about stasis and containment while music is about flow and escape. But this argument only convincing as long as you buy into its proposed divide before you debate the possible connections; if you see pattern as being the apparatus through which emotion/heart is experienced (and expressed) in music, then a head/heart divide doesn’t make much sense, for example. And where Stephen Hough sees the experience of rhythmic ‘irregularity’ as taking music away from any connection or analogy with mathematics, I suspect a mathematician might immediately think of prime numbers, and other ‘irregular’ or singular mathematical entities. And the notion of ‘unexpected’ reflects pattern-spotting competencies and experientially or culturally based perceptual expectations rather than anything intrinsically structural. Saying that music is nothing like maths because it includes unexpected developments is like saying a list of numbers is not mathematical simply because you can’t figure out (or predict) the next number in the sequence.

Say we were to ask ourselves what links between music and mathematics we could find, rather than the ways in which we could refute possible links, I think we would quickly establish that playing a musical instrument involves an exceptional degree of mathematical thinking. From spatial thinking (up, down, high, low, near, far, close, beside, under, above, and all manner of prepositional variations of ways we map and describe spatial relationships) through to fractional thinking (subdividing) through to symbolic representation of relationship, shape and direction and garden-variety counting: even when a musician is completely focussed on an emotional journey or an artistic truth, the expression of that journey and truth cannot take place without the aid of mathematical thinking.

So how do a significant number of musicians manage to persuade themselves that their music has no relationship to mathematics (if we accept that the two are deeply linked)? My first instinct and considered judgment is to blame it on poor mathematics education in primary and early secondary schools; if you don’t understand what maths is then you are unlikely to credit it as being much use or relevance to the things that define your identity.

I’ve not explored the research or analysis of that finding (what is it about early acquisition of mathematical skills and concepts that facilitates enhanced social and emotional skills?, is this a causal or a casual link?, etc.), but the idea that mathematical skillfulness has emotional and social benefits surely challenges every cliché that exists in the western educational model about maths and the limits of its purpose in education.

So far I am deeply persuaded that music and mathematics have complex connections, overlaps, correspondences and links, and the fact that we debate the existence of those links is mostly a sign of how little western culture understands what mathematics is.

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4 thoughts on “Music and Mathematics Part 1”

A well-thought out discussion, Elissa. There is passion behind your words – passion which is not normally associated with mathematics! Yet mathematicians will be passionate about their discoveries in the world of numbers, shapes and dimensions.

Indeed, at one stage in my life there were several distinguished mathematicians from around the world who sat at my dining table from time to time. I liked to ask them about how they developed those elegant ground-breaking formulas they created – were they the result of solid logical deduction, or did they received an inspiration which they then were able to justify by solid logical deduction? The answer was invariably inspiration….

Just as for us musicians there is a universe of sound which we explore, there is a universe of mathematics which underlies the world around us. There is a wonderful description in one of Oliver Sach’s books about twin idiot-savants, entertaining each other by naming successive prime numbers, in turn. They were up in the 700,000s, if I recall correctly. It was as if they only had to look in their garden of numbers to see where the primes were, like looking for forget-me-nots in a cottage garden.

Music is about vibrations of a certain kind, in a universe created out of matter which is fundamentally vibration. Mathematics is just another way of exploring the nature of the universe…expanding the mind.

But on to the broader theme… Stephen Hough would be the exception in my experience. For the most part I encounter recognition of the affinities between music and maths (perhaps I just hang around nerdy, maths-inclined people, even when it comes to musicians). I’m fascinated by stories of musicians who demonstrated mathematical or related interests. Prokofiev and his chess-playing, for example.

Yeah, agreed on the connection but usually find it hard to make it practical beyond the examples you’ve suggested, ie 1, 2, 3, 4 and repeated.

Maybe it’s just a general idea of symmetry or balance, in math you’re always trying to balance an equation, in music you could say you’re balancing a song, a note, play low sing high, play high sing low, adding new instruments (variables) but still connected, jamming.

Noticing pattern is mathematical thinking. Anything in music that involves the musician noticing pattern therefore exists in the shared territories of music and maths. And that’s without anything involving ‘greater than’, ‘lesser than’ or ‘equal to’ that occurs in music (which is nearly everything). We restrict what we think maths is to simple arithmetic and possibly dealing with geometry, and we forget that mathematics is a vastly larger field.